Geostrophic Adjustment on the Mid-Latitude β-Plane

The theory of the transition from an unbalanced initial state to a geostrophically balanced state, known as geostrophic adjustment, is a fundamental theory in geophysical fluid dynamics that originated on the f-plane in the late 1930s. Little advances were made since then in extending this theory to the mid-latitude β-plane and the present study does it for the particular initial condition of a resting fluid with step-like initial height distribution η0. In my presentation I will focus on the adjustment theory in a meridional domain of width L when η0 = η0(y) and the solutions remain zonally-invariant at all times. The effect of β on the adjustment process is examined by solving the linearized rotating shallow water equations numerically and identifying the transient waves. In the one-dimensional case, where the step in η0(y) parallels the domain's zonal walls, the meridionally propagating Poincaré waves are reflected back into the domain upon reaching the zonal walls. Thus, the geostrophic state only represents the time-averaged solution over many wave periods and not a steady-state that is actually reached by the system at long times. Our main results are: (i) The effect of β on the geostrophic state is significant only when b=cot(Φ0) Rd/a ≥0.5 (where Φ0 is the domain's central latitude, Rd is the radius of deformation and a is Earth's radius). (ii) In wide domains (e.g. L=60Rd) the effect of β on the waves is significant even for small b (e.g. b=0.005). (iii) For this small b=0.005 harmonic waves approximate the temporal evolution in narrow domains (e.g. L = 4Rd) while trapped waves approximate the temporal evolution in wide domains (e.g. L = 60Rd). In the two-dimensional case when η0 = η0(x) the transient waves propagate in the x-direction and Rossby waves are also generated. Our numerical solutions for symmetric η0 = η0(x) (a Top Hat distribution) demonstrate that the results outlined in (ii) and (iii) above also hold in this case. In addition, at short times the spatial structure of the steady solution is similar to that on the f-plane while at long times this state drifts westward at the speed of Rossby waves i.e. harmonic Rossby waves in narrow channels and trapped Rossby waves in wide channels.